Sigma chromatic number of graph coronas involving complete graphs
Agnes D. Garciano, Maria Czarina T. Lagura, Reginaldo M. Marcelo · Journal of Physics Conference Series · 2020
Abstract Let c : V(G) → ℕ be a coloring of the vertices in a graph G. For a vertex u in G, the color sum of u, denoted by σ(u), is the sum of the colors of the neighbors of u. The coloring c is called a sigma coloring of G if σ(u) ≠ σ(v) whenever u and v are adjacent vertices in G. The minimum number of colors that can be used in a sigma coloring of G is called the sigma chromatic number of G and is denoted by σ(G). Given two simple, connected graphs G and H, the corona of G and H, denoted by G ⊙ H, is the graph obtained by taking one copy of G and |V(G)| copies of H and where the ith vertex of G is adjacent to every vertex of the ith copy of H. In this study, we will show that for a graph G with |V(G)| ≥ 2, and a complete graph Kn of order n, n ≤ σ(G ⊙ Kn ) ≤ max {σ(G), n}. In addition, let Pn and Cn denote a path and a cycle of order n respectively. If m, n ≥ 3, we will prove that σ(Km ⊙ Pn ) = 2 if and only if m ≤ n − 2 ⌊ n 4 ⌋ + 2 . If n is even, we show that σ(Km ⊙ Cn ) = 2 if and only if m ≤ n − 2 ⌈ n 4 ⌉ + 2 . Furthermore, in the case that n is odd, we show that σ(Km ⊙ Cn ) = 3 if and only if m ≤ H ( ⌈ n 4 ⌉ − 1 , n − ⌈ n 4 ⌉ ) where H(r, s) denotes the number of lattice points in the convex hull of points on the plane determined by the integer parameters r and s.